Distortion elements for surface homeomorphisms

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Distortion elements for surface homeomorphisms

Emmanuel Militon

To cite this version:

Emmanuel Militon. Distortion elements for surface homeomorphisms. Geometry and Topology, Math-

ematical Sciences Publishers, 2014, 18 (1), pp.521-614. �hal-00713497v3�

Distortion elements for surface homeomorphisms

E. Militon 3 janvier 2013

Abstract

Let S be a compact orientable surface and f be an element of the group Homeo

0

(S) of homeo- morphisms of S isotopic to the identity. Denote by f ˜ a lift of f to the universal cover S ˜ of S. In this article, the following result is proved: if there exists a fundamental domain D of the covering S ˜ → S such that

n→+∞

lim 1

n d

n

log(d

n

) = 0,

where d

n

is the diameter of f ˜

ⁿ

(D), then the homeomorphism f is a distortion element of the group Homeo

0

(S).

1 Introduction

Given a compact manifold M , we denote by Diff ^r ₀ (M ) the identity component of the group of C ^r - diffeomorphisms of M . A way to understand better this group is to try to describe the subgroups of this group. In other words, given a group G, does there exist an injective group morphism from the group G to the group Diff ^r ₀ (M )? If, for this group G, we can answer affirmatively to this first question, one can try to describe the group morphisms from the group G to the group Diff ^r ₀ (M ) as best as possible (ideally up to conjugacy but this is often an unattainable goal). The concept of distortion element, which we will define, allows to obtain rigidity results on group morphisms from G to Diff ^r ₀ (M ) and will give us very partial answers to these questions.

Let us give now the definition of distortion elements. Remember that a group G is finitely generated if there exists a finite generating set G: any element g in this group is a product of elements of G and their inverses, g = s ^ǫ ₁

¹

s ^ǫ ₂

²

. . . s ^ǫ _n where the s i ’s are elements of G and the ǫ i are equal to +1 or −1. The minimal integer n in such a decomposition is denoted by l

G

(g). The map l

G

is invariant under inverses and satisfies the triangle inequality l

G

(gh) ≤ l

G

(g) + l

G

(h). Therefore, for any element g in the group G, the sequence (l

G

(g ⁿ )) n≥0 is sub-additive, so the sequence ( ^l

^G

^(g _n

ⁿ

⁾ ) n converges. When the limit of this sequence is zero, the element g is said to be distorted or a distortion element in the group G. Notice that this notion does not depend on the generating set G. In other words, this concept is intrinsic to the group G. The notion extends to the case where the group G is not finitely generated by saying that an element g of the group G is distorted if it belongs to a finitely generated subgroup of G in which it is distorted. The main interest of the notion of distortion is the following rigidity property for groups morphisms: for a group morphism ϕ : G → H , if an element g is distorted in the group G, then its image under ϕ is also distorted. In the case where the group H does not contain distortion element other than the identity element in H and where the group G contains a distortion element different from the identity, such a group morphism cannot be an embedding: the group G is not a subgroup of H .

Let us give now some simple examples of distortion elements. In any group G, the torsion elements, i.e. those of finite order, are distorted. In free groups and free abelian groups, the only distorted element is the identity element. The simplest examples of groups which contain a distortion element which is not a torsion element are the Baumslag-Solitar groups which have the following presentation : BS(1, p) =<

a, b | bab

⁻¹

= a ^p >. Then, for any integer n : b ⁿ ab

⁻ⁿ

= a ^p

ⁿ

. Taking G = {a, b} as a generating set of this

group, we have l

G

(a ^p

ⁿ

) ≤ 2n + 1 and the element a is distorted in the group BS(1, p) if |p| ≥ 2. A group G is said to be nilpotent if the sequence of subgroups (G n ) n≥0 of G defined by G 0 = G and G n+1 = [G n , G]

(this is the subgroup generated by elements of the form [g n , g] = g n gg

⁻¹

_n g

⁻¹

, where g n ∈ G n and g ∈ G) stabilizes and is equal to {1 G } for a sufficiently large n. A typical example of a nilpotent group is the Heisenberg group which is the group of upper triangular matrices whose diagonal entries are 1 and other entries are integers. In a nilpotent non-abelian group N , one can always find three distinct elements a, b and c different from the identity such that [a, b] = c and the element c commutes with a and b. In this case, we have c ⁿ

²

= [a ⁿ , b ⁿ ] so that, in the subgroup generated by a and b (and also in N), the element c is distorted: l

{a,b}

(c ⁿ

²

) ≤ 4n. A general theorem by Lubotzky, Mozes and Raghunathan implies that there exist distortion elements (and even elements with a logarithmic growth) in some lattices of higher rank Lie groups, for instance in the group SL n ( Z ) for n ≥ 3. In the case of the group SL n ( Z ), one can even find a generating set consisting of distortion elements. Notice that, in mapping class groups (see [7]) and in the group of interval exchange transformations (see [22]), any distorted element is a torsion element.

Let us consider now the case of diffeomorphisms groups. The following theorem has led to progress on Zimmer’s conjecture. Let us denote by S a compact surface without boundary endowed with a probability measure area with full support. Finally, let us denote by Diff ¹ (S, area) the group of C ¹ -diffeomorphisms of the surface S which preserve the measure area. Then we have the following statement:

Theorem. (Polterovich [23], Franks-Handel [11]) If the genus of the surface S is greater than one, any distortion element in the group Diff ¹ (S, area) is a torsion element.

As nilpotent groups and SL n (Z) have some non-torsion distortion elements, they are not subgroups of the group Diff ¹ (S, area). In the latter case, using a property of almost simplicity of the group SL n (Z), one can see even that a group morphism from the group SL n (Z) to the group Diff ¹ (S, area) is "almost"

trivial (its image is a finite group). Franks and Handel proved actually a more general result on distorsion elements in the case where the measure area is any borelian probability measure which allows them to prove that this last statement is true for any measure area with infinite support. They also obtain similar results in the cases of the torus and of the sphere. A natural question now is to wonder whether these theorems can be generalized in the case of more general diffeomorphisms or homeomorphisms groups (with no area-preservation hypothesis).

Unfortunately, one may find lots of distorted elements in those cases. The most striking example of this phenomenon is the following theorem by Calegari and Freedman:

Theorem. (Calegari-Freedman [5]) For an integer d ≥ 1, every homeomorphism in the group Homeo 0 (S ^d ) is distorted.

In the case of a higher regularity, Avila proved in [2] that any diffeomorphism in Diff

^∞

₀ (S ¹ ) for which arbitrarily large iterates are arbitrarily close to the identity in the C

^∞

-topology (such an element will be said to be recurrent) is distorted in the group Diff

^∞

₀ (S ¹ ): for instance, the irrational rotations are distorted. Using Avila’s techniques and a local perfection result (due to Haller, Rybicki and Teichmann [17]), I obtained the following result (see [20]):

Theorem 1. For any compact manifold M without boundary, any recurrent element in Diff

^∞

₀ (M ) is distorted in this group.

For instance, irrational rotations of the 2 dimensional sphere or translations of the d-dimensional torus are distorted. More generally, there exist non-trivial distortion elements in the group of C

^∞

- diffeomorphism of any manifold which admits a non-trivial C

^∞

circle action. Notice that, thanks to the Anosov-Katok method (see [16] and [8]), we can build recurrent elements in the case of the sphere or of the 2-dimensional torus which are not conjugate to a rotation. Anyway, we could not hope for a result analogous to the theorem by Polterovich and Franks and Handel, at least in the C ¹ category, as we will see that the Baumslag-Solitar group BS(1, 2) embeds in the group Diff ¹ ₀ (M ) for any manifold M (this was indicated to me by Isabelle Liousse).

Identify the circle S ¹ with R ∪{∞}. Then consider the (analytical) circle diffeomorphisms a : x 7→ x+ 1

and b : x 7→ 2x. The relation bab

⁻¹

= a ² is satisfied and, therefore, the two elements a and b define an

action of the group BS(1, 2) on the circle. By thickening the point at infinity (i.e. by replacing the point at infinity with an interval), we have a compactly-supported action of our group on R. This last action can be made C ¹ . Finally, by a radial action, we have a compactly-supported C ¹ action of this Baumslag- Solitar group on R ^d and, by identifying an open disc of a manifold with R ^d , we get an action of the Baumslag-Solitar group on any manifold. This gives some non-recurrent distortion elements in the group Diff ¹ ₀ (M ) for any manifold M . In the case of diffeomorphisms, it is difficult to approach a characterization of distortion element as there are many obstructions to being a distortion element (for instance, the differential cannot grow too fast along an orbit, the topological entropy of the diffeomorphism must vanish). On the contrary, in the groups of surface homeomorphisms, there is only one known obstruction to being a distortion element. We will describe it in the next section.

In this article, we will try to characterize geometrically the set of distortion elements in the group of homeomorphisms isotopic to the identity of a compact orientable surface. The theorem we obtain is a consequence of a result valid on any manifold and proved in the fourth section. This last result has a major drawback: it uses the fragmentation length which is not well understood except in the case of spheres.

Thus, we will try to connect this fragmentation length to a more geometric quantity: the diameter of the image of a fundamental domain under a lift of the given homeomorphism. It is not difficult to prove that the fragmentation length dominates this last quantity: this will be treated in the third section of this article. However, conversely, it is more difficult to show that this last quantity dominates the fragmentation length. In order to prove this, we will make a distinction between the case of surfaces with boundary (Section 5), which is the easiest, the case of the torus (Section 6) and the case of higher genus closed manifolds (Section 7). The last section of this article gives examples of distortion elements in the group of homeomorphisms of the annulus for which the growth of the diameter of a fundamental domain is "fast".

2 Notations and results

Let M be a manifold, possibly with boundary. We denote by Homeo 0 (M ) (respectively Homeo 0 (M, ∂M )) the identity component of the group of compactly-supported homeomorphisms of M (respectively of the group of homeomorphisms of M which pointwise fix a neighbourhood of the boundary

∂M of M ). Given two homeomorphisms f and g of M and a subset A of M , an isotopy between f and g relative to A is a continuous path of homeomorphisms (f t ) t∈[0,1] which pointwise fix A such that f 0 = f and f 1 = g. If A is the empty set, it is called an isotopy between f and g.

In what follows, S is a compact orientable surface, possibly with boundary, different from the disc and from the sphere. We denote by Π : ˜ S → S the universal cover of S. The surface S ˜ is seen as a subset of the euclidean plane R ² or of the hyperbolic plane H ² so that deck transformations are isometries for the euclidean metric or the hyperbolic metric. We endow the surface S ˜ with this metric. In what follows, we identify the fundamental group Π 1 (S) of the surface S with the group of deck transformations of the covering Π : ˜ S → S. If A is a subset of the hyperbolic plane H ² (respectively of the euclidean plane R ² ), we denote by δ(A) the diameter of A for the hyperbolic distance (respectively the euclidean distance).

For a homeomorphism f of S, a lift of f is a homeomorphism F of S ˜ which satisfies Π ◦ F = f ◦ Π.

For an isotopy (f t ) _t∈[0,1] , a lift of (f t ) _t∈[0,1] is a continuous path (F t ) _t∈[0,1] of homeomorphisms of S ˜ such that, for any t, the homeomorphism F t is a lift of the homeomorphism f t . For a homeomorphism f in Homeo 0 (S), we denote by f ˜ a lift of f obtained as the time 1 of a lift of an isotopy between the identity and f which is equal to the identity for t = 0. If moreover the boundary of S is non-empty and the homeomorphism f is in Homeo 0 (S, ∂S), the homeomorphism f ˜ is obtained by lifting an isotopy relative to the boundary ∂S. If there exists a disc D ² embedded in the surface S which contains the support of the homeomorphism f , we require moreover that the support of f ˜ is contained in Π

⁻¹

(D ² ). Notice that the homeomorphism f ˜ is unique except in the cases of the groups Homeo 0 (T ² ) and Homeo 0 ([0, 1] × S ¹ ).

This last claim is a consequence of a theorem by Hamstrom (see [13]): if S is a surface without boundary

of genus greater than 1, then the topological space Homeo 0 (S ) is simply connected. Moreover, if S is a

surface with a nonempty boundary, the topological space Homeo 0 (S, ∂S) is simply connected. Finally,

let us prove the claim in the case of an element f in Homeo 0 (S). Take two lifts F 1 and F 2 of f to S ˜ as

above. Then the double of F 1 , which is the homeomorphism on the double of S ˜ canonically defined by F 1 , is equal to the double of F 2 by Hamstrom’s theorem. This proves the claim.

Notice that, for any deck transformation γ ∈ Π 1 (S), and any homeomorphism f in Homeo 0 (S), γ ◦ f ˜ = ˜ f ◦ γ. This is true when the surface S is the torus or the closed annulus and, in the other cases, the map γ ◦ f ˜ as well as the map f ˜ ◦ γ is the time 1 of a lift of an isotopy between the identity and f which is equal to γ for t = 0.

Definition 2.1. We call fundamental domain of S any compact connected subset D of S ˜ which satisfies the following properties:

1. Π(D) = S.

2. For any deck transformation γ in Π 1 (S) different from the identity, the interior of D is disjoint from the interior of γ(D).

The main theorem of this article is a partial converse to the following property (observed by Franks and Handel in [11], Lemma 6.1):

Proposition 2.1. Denote by D a fundamental domain of S ˜ for the action of Π 1 (S ).

If a homeomorphism f in Homeo 0 (S) (respectively in Homeo 0 (S, ∂S)) is a distortion element of Homeo 0 (S) (respectively of Homeo 0 (S, ∂S)), then:

n→+∞ lim

δ( ˜ f ⁿ (D)) n = 0.

Remark In the case where the surface considered is the torus T ² or the annulus [0, 1] × S ¹ , the conclusion of this proposition is equivalent to saying that the rotation set of f is reduced to a single point (see [21]

for a definition of the rotation set of a homeomorphism of the torus isotopic to the identity; the definition is analogous in the case of the annulus).

Proof. Let f be a distortion element in Homeo 0 (S) (respectively in Homeo 0 (S, ∂S)). Denote by G = {g 1 , g 2 , . . . , g p } a finite subset of Homeo 0 (S) (respectively of Homeo 0 (S, ∂S)) such that:

1. The homeomorphism f belongs to the group generated by G.

2. The sequence ( ^l

^G

^(f _n

ⁿ

⁾ ) n≥1 converges to 0.

Then we have a decomposition of the form:

f ⁿ = g i

1

◦ g i

2

◦ . . . ◦ g i

_ln

where l n = l

G

(f ⁿ ). This implies the following equality:

I ◦ f ˜ ⁿ = ˜ g i

1

◦ ˜ g i

2

◦ . . . ◦ g ˜ i

_ln

where I is an isometry of S. Let us take ˜ µ = max _{1≤i≤p, x∈} S ˜ d(x, g ˜ i (x)). As, for any index i and any deck transformation γ in Π 1 (S), γ ◦ ˜ g i = ˜ g i ◦ γ and as the distance d is invariant under deck transformations, µ is finite. Then, for any two points x and y of the fundamental domain D, we have:

d( ˜ f ⁿ (x), f ˜ ⁿ (y)) = d(I ◦ f ˜ ⁿ (x), I ◦ f ˜ ⁿ (y))

≤ d(I ◦ f ˜ ⁿ (x), x) + d(x, y) + d(I ◦ f ˜ ⁿ (y), y)

≤ l n µ + δ(D) + l n µ

which implies the proposition, by sublinearity of the sequence (l n ) n≥0 . The main theorem of this article is the following:

Theorem 2.2. Let f be a homeomorphism in Homeo 0 (S) (respectively in Homeo 0 (S, ∂S)). If:

n→+∞ lim

δ( ˜ f ⁿ (D))log(δ( ˜ f ⁿ (D)))

n = 0,

then f is a distortion element in Homeo 0 (S) (respectively in Homeo 0 (S, ∂S)).

Remark The property lim n→+∞ δ( ˜ f

ⁿ

(D))log(δ( ˜ f

ⁿ

(D)))

n = 0 is independent of the chosen fundamental do- main D, as we will see in the next section. Thus, it is invariant under conjugation.

The proof of this theorem occupies the next five sections. Let us now introduce a new notion in order to complete this proof.

Let M be a compact d-dimensional manifold. We will call closed ball of M the image of the closed unit ball under an embedding from R ^d to the manifold M . Let:

H ^d =

(x 1 , x 2 , . . . , x d ) ∈ R ^N , x 1 ≥ 0 .

We will call closed half-ball of M the image of B(0, 1) ∩ H ^d under an embedding p : H ^d → M such that:

p(∂H ^d ) = p(H ^d ) ∩ ∂M.

Let us fix a finite family U of closed balls or closed half-balls whose interiors cover M . Then, by the frag- mentation lemma (see [9] or [4]), there exists a finite family (f i ) 1≤i≤n of homeomorphisms in Homeo 0 (M ), each supported in one of the sets of U , such that:

f = f 1 ◦ f 2 ◦ . . . ◦ f n .

We denote by Frag

_U

(f ) the minimal integer n in such a decomposition: it is the minimal number of factors necessary to write f as a product (i.e. composition) of homeomorphisms supported each in one of the balls of U .

Let us come back to the case of a compact surface S and denote by U a finite family of closed discs or of closed half-discs whose interiors cover S. Denote by D a fundamental domain of S ˜ for the action of Π 1 (S). We now describe the two steps of the proof of Theorem 2.2. The first step of the proof consists of checking that the quantity Frag

_U

(f ) is almost equal to δ( ˜ f (D)):

Theorem 2.3. There exist two real constants C > 0 and C

^′

such that, for any homeomorphism g in Homeo 0 (S):

1

C δ(˜ g(D)) − C

^′

≤ Frag

_U

(g) ≤ Cδ(˜ g(D)) + C

^′

.

In the case when the boundary of the surface S is nonempty, let us denote by S

^′

a submanifold of S homeomorphic to S, contained in the interior of S and which is a deformation retract of S. We denote by U a family of closed balls of S whose union of interiors cover S

^′

.

Theorem 2.4. There exist two real constants C > 0 and C

^′

such that, for any homeomorphism g in Homeo 0 (S, ∂S) supported in S

^′

:

1

C δ(˜ g(D)) − C

^′

≤ Frag

_U

(g) ≤ Cδ(˜ g(D)) + C

^′

.

The lower bound of the fragmentation length is not difficult: it is treated in the next section in which we will also see that the quantity Frag

_U

is essentially independent from the cover U chosen. On the other hand, it is much more technical to establish the upper bound. In the proof of this bound, we distinguish three cases: the case of surfaces with boundary (Section 5), the case of the torus (Section 6) and the case of higher genus compact surfaces without boundary (Section 7). The proof seems to depend strongly on the fundamental group of the surface considered. In particular, it is easier in the case of surfaces with boundary whose fundamental groups are free. In the case of the torus, the proof is a little tricky and, in the case of higher genus closed surfaces, the proof is more complex and uses Dehn algorithm for small-cancellation groups (surface groups in this case).

Let us explain now the second step of the proof. Denote by M a compact manifold and U a finite

family of closed balls or half-balls whose interiors cover M . In Section 4, we will prove the following

theorem which asserts that, for a homeomorphism f in Homeo 0 (M ), if the sequence Frag

_U

(f ⁿ ) does not

grow too fast with n, then the homeomorphism f is a distortion element:

Theorem 2.5. If

n→+∞ lim

Frag

_U

(f ⁿ ).log(Frag

_U

(f ⁿ ))

n = 0,

then the homeomorphism f is a distortion element in Homeo 0 (M ).

Moreover, in the case of a manifold M with boundary, if U is a finite family of closed balls contained in the interior of M whose interiors cover the support of a homeomorphism f in Homeo 0 (M, ∂M ), this last theorem remains true in the group Homeo 0 (M, ∂M ). This section uses a technique due to Avila (see [2]).

Theorem 2.2 is clearly a consequence of these two theorems.

The last section is dedicated to the proof of the following theorem which proves that Proposition 2.1 is optimal:

Theorem 2.6. Let (v n ) n≥1 be a sequence of positive real numbers such that:

n→+∞ lim v n

n = 0.

Then there exists a homeomorphism f in Homeo 0 (R/Z × [0, 1], R/Z × {0, 1}) such that:

1. ∀n ≥ 1, δ( ˜ f ⁿ ([0, 1] × [0, 1])) ≥ v n .

2. The homeomorphism f is a distortion element in Homeo 0 (R/Z × [0, 1], R/Z × {0, 1}).

This theorem means that being a distortion element gives no clues on the growth of the diameter of a fundamental domain other than the sublinearity of this growth. This theorem remains true for any surface S: it suffices to embed the annulus R/Z × [0, 1] in any surface S and to use this last theorem to see it.

3 Quasi-isometries

In this section, we prove the lower bound in Theorems 2.3 and 2.4. More precisely, we prove these theorems using the following propositions whose proof will be discussed in Sections 5, 6 and 7.

Proposition 3.1. There exists a finite cover U of S by closed discs and half-discs as well as real constants C ≥ 1 and C

^′

≥ 0 such that, for any homeomorphism g in Homeo 0 (S):

Frag

_U

(g) ≤ Cdiam

D

(˜ g(D 0 )) + C

^′

.

Here is a version of the previous proposition in the case of the group Homeo 0 (S, ∂S).

Proposition 3.2. Fix a subsurface with boundary S

^′

of S which is contained in the interior of S, is a deformation retract of S and is homeomorphic to S. There exists a finite cover U of S

^′

by closed discs contained in the interior of S as well as real constants C ≥ 1 and C

^′

≥ 0 such that, for any homeomorphism g in Homeo 0 (S) supported in S

^′

:

Frag

_U

(g) ≤ Cdiam

D

(˜ g(D 0 )) + C

^′

.

In order to prove these theorems, we need some notation. As in the last section, let us denote by S a compact surface. Two maps a, b : Homeo 0 (S) → R are quasi-isometric if and only if there exist real constants C ≥ 1 and C

^′

≥ 0 such that:

∀f ∈ Homeo 0 (S), 1

C a(f ) − C

^′

≤ b(f ) ≤ Ca(f ) + C

^′

.

More generally, an arbitrary number of maps Homeo 0 (S) → R are said to be quasi-isometric if they are pairwise quasi-isometric.

Let us consider a fundamental domain D 0 of S ˜ for the action of the group Π 1 (S) which satisfies the following properties (see figure 1) :

1. If the surface S is closed of genus g, the fundamental domain D 0 is a closed disc bounded by a 4g-gone with geodesic edges.

2. If the surface S has a nonempty boundary, the fundamental domain D 0 is a closed disc bounded by a polygon with geodesic edges such that any edge of this polygon which is not contained in ∂ S ˜ connects two edges contained in ∂ S. ˜

∂ S ˜

∂ S ˜ ∂ S ˜

∂ S ˜

Case of the torus Case of the torus with one hole Case of the genus 2 closed surface

D

₀

D

₀

D

₀

Figure 1: The fundamental domain D 0

Let us take:

D = {γ(D 0 ), γ ∈ Π 1 (S)} .

For fundamental domains D and D

^′

in D, we denote by d

D

(D, D

^′

)+ 1 the minimal number of fundamental domains met by a path which connects the interior of D to the interior of D

^′

. The map d

D

is a distance on D. We now give an algebraic definition of this quantity. Denote by G the finite set of deck transformations in Π 1 (S) which send D 0 to a polygon in D adjacent to D 0 , i.e. which shares an edge in common with D 0 . Then the subset G is symmetric and is a generating set of Π 1 (S). Notice that the map

d

G

: Π 1 (S) × Π 1 (S) → R (ϕ, ψ) 7→ l

G

(ϕ

⁻¹

ψ)

is a distance on the group Π 1 (S). Then, for any pair (ϕ, ψ) of deck transformations in the group Π 1 (S), we have:

l

G

(ϕ

⁻¹

ψ) = d

D

(ϕ(D 0 ), ψ(D 0 )).

One can see it by noticing that d

D

is invariant under the action of the group Π 1 (S) and by proving by induction on l

G

(ψ) that:

l

G

(ψ) = d

D

(D 0 , ψ(D 0 )).

Given a compact subset A of S, we call ˜ discrete diameter of A the following quantity:

diam

D

(A) = max

d

D

(D, D

^′

),

D ∈ D, D

^′

∈ D D ∩ A 6= ∅, D

^′

∩ A 6= ∅

.

For a fundamental domain D 1 in D, we call éloignement of A with respect to D 1 the following quantity:

el D

1

(A) = max

d

D

(D 1 , D),

D ∈ D D ∩ A 6= ∅

.

Notice that, in the case where D 1 ∩ A 6= ∅, we have:

el D

1

(A) ≤ diam

D

(A) ≤ 2el D

1

(A).

In this section, we prove the following statement, using Proposition 3.1:

Proposition 3.3. Given two finite sets U and U

^′

of closed balls or half-balls whose interiors cover the surface S, the maps Frag

_U

and Frag

_U^′

are quasi-isometric.

Given two fundamental domains D and D

^′

of S ˜ for the action of the fundamental group of S , the maps f 7→ δ( ˜ f (D)) and g 7→ δ(˜ g(D

^′

)) are quasi-isometric.

Fix now a finite cover U of S as above and a fundamental domain D. Then the following maps Homeo 0 (S) → R are quasi-isometric:

1. The map Frag

_U

. 2. The map g 7→ δ(˜ g(D)).

3. The map g 7→ diam

D

(˜ g(D 0 )).

When the boundary of the surface S is nonempty, we have an analogous proposition in the case of the group Homeo 0 (S, ∂S). As in the last section, let us denote by S

^′

a submanifold with boundary of S homeomorphic to S, contained in the interior of S, and which is a deformation retract of S, and by U a finite family of closed balls contained in the interior of S whose union of the interiors contains S

^′

. Finally, let us denote by G S

^′

the group of homeomorphisms in Homeo 0 (S, ∂S) which are supported in S

^′

. Proposition 3.4. Let us fix a fundamental domain D of S ˜ for the action of the fundamental group of S.

The following maps G S

^′

→ R are quasi-isometric:

1. The map Frag

_U

. 2. The map g 7→ δ(˜ g(D)).

3. The map g 7→ diam

D

(˜ g(D 0 )).

The proof of this proposition is quite the same as the proof of the previous one: that is why we will not provide it.

These two propositions directly imply Theorems 2.3 and 2.4.

Proof. Let us prove first that, for any two fundamental domains D and D

^′

, the maps g 7→ δ(˜ g(D)) and g 7→ δ(˜ g(D

^′

)) are quasi-isometric. Let us take:

{γ 1 , γ 2 , . . . , γ p } = {γ ∈ Π 1 (S), D

^′

∩ γ(D) 6= ∅} . Notice that:

D

^′

⊂

p

[

i=1

γ i (D) and the right-hand side is path-connected. Then:

˜ g(D

^′

) ⊂

p

[

i=1

˜

g(γ i (D)).

Then the lemma below implies that:

δ(˜ g(D

^′

)) ≤ pδ(˜ g(D)).

As the fundamental domains D and D

^′

play symmetric roles, this implies that the maps g 7→ δ(˜ g(D)) and g 7→ δ(˜ g(D

^′

)) are quasi-isometric.

Lemma 3.5. Let X be a path-connected metric space. Let (A i ) 1≤i≤p be a family of closed subsets of X such that:

X =

p

[

i=1

A i .

Then:

δ(X ) = sup

x∈X,y∈X

d(x, y) ≤ p max

1≤i≤p δ(A i ).

Proof. Let x and y be two points in X. By path-connectedness of X , there exists an integer k between 1 and p, an injection σ : [1, k] ∩ Z → [1, p] ∩ Z and a sequence (x i ) 1≤i≤k+1 of points in X which satisfy the following properties:

1. x 1 = x and x k+1 = y.

2. For any index i between 1 and k, the points x i and x i+1 both belong to A σ(i) . Then:

d(x, y) ≤ P ^k

i=1

d(x i , x i+1 )

≤ P ^k

i=1

δ(A σ(i) )

≤ p max

1≤i≤p δ(A i ).

This last inequality implies the lemma.

Let us show now that, for two finite families U and U

^′

as in the statement of Proposition 3.3, the maps Frag

_U

and Frag

_U^′

are quasi-isometric. The proof of this fact requires the following lemmas.

Lemma 3.6. Let ǫ > 0. Let us denote by B the unit closed ball of R ^d . There exists an integer N ≥ 0 such that any homeomorphism in Homeo 0 (B, ∂B) can be written as a composition of at most N homeo- morphisms in Homeo 0 (B, ∂B) ǫ-close to the identity (for a distance which defines the C ⁰ -topology on this group).

Lemma 3.7. Let M be a compact manifold and {U 1 , U 2 , . . . , U p } be an open cover of M . There ex- ist ǫ > 0 and an integer N

^′

> 0 such that, for any homeomorphism g in Homeo 0 (M ) (respectively in Homeo 0 (M, ∂M )) ǫ-close to the identity, there exist homeomorphisms g 1 , . . . , g N

^′

in Homeo 0 (M ) (respec- tively in Homeo 0 (M, ∂M )) such that:

1. Each homeomorphism g i is supported in one of the U j ’s.

2. g = g 1 ◦ g 2 ◦ . . . ◦ g N

^′

.

Lemma 3.6 is a consequence of Lemma 5.2 in [3] (notice that the proof works in dimensions higher than 2). Lemma 3.7 is classical. It is a consequence of the proof of Theorem 1.2.3 in [4]. These two lemmas imply that, for an open cover of the disc D ² , there exists an integer N such that any homeomorphism in Homeo 0 (D ² , ∂D ² ) can be written as a composition of at most N homeomorphisms supported each in one of the open sets of the covering. Now, for an element U in U , we denote by U ∩ U

^′

the cover of U given by the intersections of the elements of U

^′

with U . The application of this last result to the ball U with the cover U ∩ U

^′

gives us a constant N U . Let us denote by N the maximum of the N U , where U varies over U . We directly obtain that, for any homeomorphism g:

Frag

_U^′

(g) ≤ NFrag

_U

(g).

As the two covers U and U

^′

play symmetric roles, the fact is proved. Notice that this fact is true in any dimension.

Using a quasi-isometry between the metric spaces (Π 1 (S), d S ) and S, we will prove the following lemma ˜ which implies that the last two maps in the proposition are quasi-isometric:

Lemma 3.8. There exist constants C ≥ 1 and C

^′

≥ 0 such that, for any compact subset A of S: ˜ 1

C δ(A) − C

^′

≤ diam

D

(A) ≤ Cδ(A) + C

^′

. Proof. Let us fix a point x 0 in the interior of D 0 . The map:

q : Π 1 (S) → S ˜

γ 7→ γ(x 0 )

is a quasi-isometry for the distance d

G

and the distance on S ˜ (this is the Švarc-Milnor lemma, see [14]

p.87). We notice that, for a compact subset A of S, the number ˜ diam

D

(A) is equal to the diameter of q

⁻¹

(B) for the distance d

G

, where

B = [

{D, D ∈ D D ∩ A 6= ∅} .

We deduce that there exist constants C 1 ≥ 1 and C ₁

^′

≥ 0 independent from A such that:

1

C 1 δ(B) − C ₁

^′

≤ diam

D

(A) ≤ C 1 δ(B) + C ₁

^′

. The inequalities

δ(B) − 2δ(D 0 ) ≤ δ(A) ≤ δ(B), complete the proof of the lemma.

We now prove that, for any cover U as in the statement of Proposition 3.3, there exist constants C ≥ 1 and C

^′

≥ 0 such that, for any homeomorphism g in Homeo 0 (S):

1

C diam

D

(˜ g(D 0 )) − C

^′

≤ Frag

_U

(g).

Let us fix such a family U . We will need the following lemma that we will prove later:

Lemma 3.9. There exists a constant C > 0 such that, for any compact subset A of S ˜ and any homeo- morphism g supported in one of the sets in U :

diam

D

(˜ g(A)) ≥ diam

D

(A) − C.

Take k = Frag

_U

(g) and:

g = g 1 ◦ g 2 ◦ . . . ◦ g k ,

where each homeomorphism g i is supported in one of the elements of U . Then:

I ◦ g ˜ = ˜ g 1 ◦ ˜ g 2 ◦ . . . ◦ g ˜ k ,

where I is a deck transformation (and an isometry). Lemma 3.9 combined with an induction implies that:

∀j ∈ [1, k] ∩ Z , diam

D

(˜ g _j

⁻¹

◦ . . . ◦ ˜ g

⁻¹

₁ ◦ ˜ g(D 0 )) ≥ diam

D

(˜ g(D 0 )) − jC, as the homeomorphisms g ˜ i commute with I. Hence:

2 = diam

D

(˜ g _k

⁻¹

◦ . . . ◦ ˜ g

⁻¹

₁ ◦ g(D ˜ 0 )) ≥ diam

D

(˜ g(D 0 )) − kC.

Therefore:

Frag

_U

(g) ≥ 1

C diam

D

(˜ g(D 0 )) − 2 C . We obtain the lower bound wanted.

Proof of Lemma 3.9. For an element U in U , we denote by U ˜ a lift of U , i.e. a connected component of Π

⁻¹

(U ). Let us take:

µ(U ) = diam

D

( ˜ U).

This quantity does not depend on the lift U ˜ chosen. We denote by µ the maximum of the µ(U ), for U in U .

We denote by U g an element in U which contains the support of g. Let us consider two fundamental domains D and D

^′

which meet A and which satisfy the following relation:

d

D

(D, D

^′

) = diam

D

(A).

Let us take a point x in D ∩ A and a point x

^′

in D

^′

∩ A. If the point x belongs to Π

⁻¹

(U g ), we denote by U ˜ g the lift of U g which contains x. Then the point ˜ g(x) belongs to U ˜ g and a fundamental domain D ˆ which contains the point g(x) ˜ is at distance at most µ from D (for d

D

). Hence, in any case, there exists a fundamental domain D ˆ which contains the point ˜ g(x) and is at distance at most µ from D. Similarly, there exists a fundamental domain D ˆ

^′

which contains the point ˜ g(x

^′

) and is at distance at most µ from D

^′

. Therefore:

d

D

( ˆ D, D ˆ

^′

) ≥ d

D

(D, D

^′

) − 2µ.

We deduce that:

diam

D

(˜ g(A)) ≥ diam

D

(A) − 2µ, what we wanted to prove.

Thus, to complete the proof of Proposition 3.3, it suffices to prove Proposition 3.1.

It suffices now to find a finite family U for which Proposition 3.1 or 3.2 holds. We will distinguish the following cases. A section is devoted to each of them:

1. The surface S has a nonempty boundary (Section 5).

2. The surface S is the torus (Section 6).

3. The surface S is closed of genus greater than one (Section 7).

The proof of Propositions 3.1 and 3.2, in each of these cases, consists in putting back the boundary of

˜

g(D 0 ) close to the boundary of ∂D 0 by composing by homeomorphisms supported each in the interior of one of the balls of a well-chosen cover U . Most of the time, after composing by a homeomorphism supported in the interior of one of the balls of U , the image of the fundamental domain D 0 will not meet faces which were not met before the composition. However, it will not be always possible, which explains the technicality of parts of the proof. Then, we will have to assure that, after composing by a uniformly bounded number of homeomorphisms supported in interiors of balls of U , the image of the boundary of D 0 will be strictly closer to D 0 than before.

4 Distortion and fragmentation on manifolds

In this section, M is a compact d-dimensional manifold, possibly with boundary. Let us fix a finite family U of closed balls or half-balls of M whose interiors cover M . For a homeomorphism g in Homeo 0 (M ), we denote by a

U

(g) the minimum of the quantities l.log(k), where there exists a finite set {f _i , 1 ≤ i ≤ k} of k homeomorphisms in Homeo 0 (M ), each supported in one of the elements of U , and a map ν : [1, l]∩ Z >0 → [1, k] ∩ Z >0 with:

g = f ν(1) ◦ f ν(2) ◦ . . . ◦ f ν(l) . The aim of this section is to prove the following proposition:

Proposition 4.1. Let f be a homeomorphism in Homeo 0 (M ). Then:

n→+∞ lim

a

U

(f ⁿ ) n = 0

if and only if the homeomorphism f is a distortion element in Homeo 0 (M ).

Let us give now an analogous statement in the case of the group Homeo 0 (M, ∂M ). Denote by M

^′

a submanifold with boundary of M homeomorphic to M , contained in the interior of M and which is a

deformation retract of M . We denote by U a family of closed balls of M whose interiors cover M

^′

. For a

homeomorphism g in Homeo 0 (M, ∂M ) supported in M

^′

, we define a

U

(g) the same way as before.

Proposition 4.2. Let f be a homeomorphism in Homeo 0 (M, ∂M ) supported in M

^′

. Then:

n→+∞ lim

a

U

(f ⁿ ) n = 0

if and only if the homeomorphism f is a distortion element in Homeo 0 (M, ∂M ).

As a

U

(f ) ≤ Frag

_U

(f ).log(Frag

_U

(f )), these last propositions clearly imply Theorem 2.5.

Proof of the "if" statement in Propositions 4.1 and 4.2. If the homeomorphism f is a distortion element, we denote by G the finite set which appears in the definition of a distortion element. Then we write each of the homeomorphisms in G as a product of homeomorphisms supported in one of the sets of U . We denote by G

^′

the (finite) set of homeomorphisms which appear in such a decomposition. Then the homeomorphism f ⁿ is a composition of l n elements of G

^′

, where l n is less than a constant independent from n times l

G

(f ⁿ ). As the element f is distorted, lim n→+∞ l

n

n = 0 and:

a

U

(f ⁿ ) ≤ log(card(G

^′

))l n . Therefore:

n→+∞ lim

a

U

(f ⁿ ) n = 0.

In the case of Proposition 4.2, there is only one new difficulty: the elements of G are not necessarily supported in the union of the balls of U . Let us take a homeomorphism h in Homeo 0 (M, ∂M ) with the following properties: the homeomorphism h is equal to the identity on M

^′

and sends the union of the supports of elements of G in the union of the interiors of the balls of U. Then it suffices to consider the finite set hGh

⁻¹

instead of G in order to complete the proof.

The full power of Propositions 4.1 and 4.2 will be used only for the proof of Theorem 2.6 (construction of the example). In order to prove Theorem 2.2, we just used Theorem 2.5 which is weaker.

Remark Notice that, if U is the cover of the sphere by two neighbourhoods of the hemispheres, the map Frag

_U

is bounded by 3 on the group Homeo 0 (S ⁿ ) of homeomorphisms of the n-dimensional sphere isotopic to the identity (see [5]). This is a consequence of the annulus theorem by Kirby (see [18]) and Quinn (see [24]). Thus, the following theorem by Calegari and Freedman (see [5]) is a consequence of Theorem 2.5:

Theorem 4.3 (Calegari-Freedman [5]) . Any homeomorphism in Homeo 0 (S ⁿ ) is a distortion element.

The proof of Proposition 4.1 is based on the following lemma, whose proof uses a technique due to Avila (see [2]):

Lemma 4.4. Let (f n ) n≥1 be a sequence of homeomorphisms of R ^d (respectively of H ^d ) supported in B(0, 1) (respectively in B(0, 1) ∩ H ^d ). There exists a finite set G of compactly-supported homeomorphisms of R ^d (respectively of H ^d ) such that:

1. For any natural number n, the homeomorphism f n belongs to the group generated by G.

2. l

G

(f n ) ≤ 14.log(n) + 14.

This lemma is not true anymore in case of the C ^r regularity, for r ≥ 1. It crucially uses the following fact: given a sequence of homeomorphisms (h n ) supported in the unit ball B(0, 1), one can store all the information of this sequence in one homeomorphism. Let us explain now how to build such a homeomor- phism. For any integer n, denote by g n a homeomorphism which sends the unit ball on a ball B n such that the balls B n are pairwise disjoint and have a diameter which converges to 0. Then it suffices to consider the homeomorphism

∞

Y

n=1

g n h n g

⁻¹

_n .

Such a construction is not possible in the case of a higher regularity.

Remark There are two main differences between this lemma and the one stated by Avila:

1. Avila’s lemma deals with a sequence of diffeomorphisms which converges sufficiently fast (in the C

^∞

-topology) to the identity whereas any sequence of homeomorphisms is considered here.

2. The upper bound is logarithmic and not linear.

Remark This lemma is optimal in the sense that, if the homeomorphisms f n are pairwise distinct, the growth of l

G

(f n ) is at least logarithmic. Indeed, if the generating set G contains k elements, there are at most ^k

^l+1

_k−1

⁻¹

homeomorphisms whose length with respect to G is less than or equal to l.

Before proving Lemma 4.4, let us see why this lemma implies Propositions 4.1 and 4.2.

End of the proof of Propositions 4.1 and 4.2. Suppose that:

n→+∞ lim

a

U

(f ⁿ ) n = 0.

Let

U = {U 1 , U 2 , . . . , U p } .

For any integer i between 1 and p, denote by ϕ i an embedding of R ^d into M which sends the closed ball B(0, 1) onto U i if U i is a closed ball or an embedding of H ^d into M which sends the closed half-ball B(0, 1) ∩ H ^d onto U i if U i is a closed half-ball. For any natural number n, let l n and k n be two positive integers such that:

1. a

U

(f ⁿ ) = l n log(k n ).

2. There exists a sequence (f 1,n , f 2,n , . . . , f k

n

,n ) of homeomorphisms in Homeo 0 (M ), each supported in one of the elements of U , such that f ⁿ is the composition of l n homeomorphisms of this family.

Let us build an increasing one-to-one function σ : Z >0 → Z >0 which satisfies:

∀n ∈ Z >0 , l σ(n) (14.log( P n

i=1 k σ(i) ) + 14)

σ(n) ≤ 1

n . Suppose that, for some m ≥ 0, σ(1), σ(2), . . . , σ(m) have been built. Then, as:

n→+∞ lim

l n log(k n ) n = 0, we have

n→+∞ lim

l n (14.log( P m

i=1 k _σ(i) + k n ) + 14)

n = 0.

Hence, we can find an integer σ(m + 1) > σ(m) such that:

l σ(m+1) (14.log( P m+1

i=1 k σ(i) ) + 14)

σ(m + 1) ≤ 1

m . This completes the construction of the map σ. Take a bijective map:

ψ : Z >0 →

(i, σ(j)) ∈ Z >0 × Z >0 ,

i ≤ k σ(j)

j ∈ Z >0

such that, if ψ(n 1 ) = (i 1 , σ(j 1 )), ψ(n 2 ) = (i 2 , σ(j 2 )) and σ(j 1 ) < σ(j 2 ), then n 1 < n 2 . For instance, take the inverse of the bijective map

(i, σ(j)) ∈ Z >0 × Z >0 ,

i ≤ k σ(j)

j ∈ Z >0

→ Z >0

(i, σ(j)) 7→ i + P

j

^′

<j

k _σ(j

^′

₎ .

Then:

ψ

⁻¹

(i, σ(j)) ≤

j

X

l=1

k σ(l) .

Denote by τ i,j an integer between 1 and p such that:

supp(f i,j ) ⊂ U τ

i,j

. Then apply Lemma 4.4 to the sequence of homeomorphisms

ϕ

⁻¹

_τ

_ψ(n)

◦ f ψ(n) ◦ ϕ τ

ψ(n)

,

where the ϕ i ’s were defined at the beginning of the proof. Let us denote by G the finite set given by Lemma 4.4. Let G i be the finite set of homeomorphisms supported in U i of the form ϕ i ◦ s ◦ ϕ

⁻¹

_i , where s is a homeomorphism in G. Let

G

^′

=

p

[

i=1

G i .

By Lemma 4.4:

∀n ∈ Z >0 , l

G^′

(f ψ(n) ) ≤ Clog(n) + C

^′

. Now the homeomorphism f ^σ(n) can be decomposed as follows:

f ^σ(n) = g 1 ◦ g 2 ◦ . . . ◦ g l

σ(n)

, where each of the homeomorphisms g i belongs to the set:

n f 1,σ(n) , f 2,σ(n) , . . . , f _k

_σ(n)

,σ(n)

o .

Thus:

l

G^′

(f ^σ(n) ) ≤ l σ(n) (Clog( max

1≤i≤k

σ(n)

ψ

⁻¹

(i, σ(n))) + C

^′

).

Therefore:

l

G^′

(f ^σ(n) )

σ(n) ≤ l σ(n) (C.log( P n

i=1 k σ(i) ) + C

^′

)

σ(n) ≤ 1

n

and the homeomorphism f is a distortion element of Homeo 0 (M ) (respectively of Homeo 0 (M, ∂M )).

Let us now prove Lemma 4.4. This will require two lemmas.

Let a and b be the generators of the free semigroup L 2 on two generators. For two compactly sup- ported homeomorphisms f and g of R ^d , let η f,g be the semigroup morphism from L 2 to the group of homeomorphism of R ^d defined by η f,g (a) = f and η f,g (b) = g.

Lemma 4.5. There exist compactly supported homeomorphisms s 1 and s 2 of R ^d such that:

∀m ∈ L 2 , m

^′

∈ L 2 , m 6= m

^′

⇒ η s

1

,s

2

(m)(B(0, 2)) ∩ η s

1

,s

2

(m

^′

)(B(0, 2)) = ∅ and the diameter of η s

1

,s

2

(m)(B(0, 2)) converges to 0 when the length of m tends to infinity.

Lemma 4.6. Let f be a homeomorphism in Homeo 0 (R ^d ). There exist two homeomorphisms g and h in Homeo 0 (R ^d ) such that:

f = [g, h], where [g, h] = g ◦ h ◦ g

⁻¹

◦ h

⁻¹

.

This lemma is classical and seems to appear for the first time in [1]. Let us prove it now.

Proof. Denote by ϕ a homeomorphism in Homeo 0 (R ^d ) whose restriction to B(0, 2) is defined by:

B(0, 2) → R ^d

x 7→ ^x ₂ .

s

₁

s

₂

s

₂

s

₂

s

₂

s

₂

s

₂

s

₂

s

₁

s

₁

s

₁

s

₁

s

₁

s

₁

B(0, 2)

Figure 2: Lemma 4.5

For any natural number n, let

A n =

x ∈ R ^d , 1

2 ⁿ⁺¹ ≤ kxk ≤ 1 2 ⁿ

.

Let f be an element in Homeo 0 (R ^N ). As any element in Homeo 0 (R ^N ) is conjugate to an element supported in the interior of A 0 , we may suppose that the homeomorphism f is supported in the interior of A 0 . Then we define g ∈ Homeo 0 (R ^d ) by:

1. g = Id outside B(0, 1).

2. For any natural number i, g

|Ai

= ϕ ⁱ f ϕ

⁻ⁱ

. 3. g(0) = 0.

Then:

f = [g, ϕ].

ϕ ϕ ϕ A

0

A

1

A

2

Figure 3: Proof of Lemma 4.6 : description of the homeomorphism ϕ

These two lemmas remain true when we replace R ^d with H ^d and B(0, 2) with B(0, 2) ∩ H ^d .

Before proving Lemma 4.5, let us prove Lemma 4.4 with the help of these two lemmas.

Proof of Lemma 4.4. We prove the lemma in the case of homeomorphisms of R ^d . In the case of the half- space, the proof can be performed likewise. For an element m in L 2 , let l(m) be the length of m as a word in a and b. Let

Z >0 → L 2

n 7→ m n

be a bijective map which satisfies:

l(m n ) < l(m n

^′

) ⇒ n < n

^′

. This last condition implies that:

l(m n ) = l ⇔ 2 ^l ≤ n < 2 ^l+1 . In particular, for any natural number n:

l(m n ) ≤ log 2 (n).

Let s 1 and s 2 be the homeomorphisms in Homeo 0 (R ^d ) given by Lemma 4.5. Let s 3 be a homeomorphism in Homeo 0 ( R ^d ) supported in the ball B(0, 2) which satisfies:

s 3 (B (0, 1)) ∩ B(0, 1) = ∅.

We denote by B n the closed ball η s

1

,s

2

(m n )(B(0, 1)). By Lemma 4.6, there exist homeomorphisms g n and h n supported in B (0, 1) such that f n = [g n , h n ].

... ...

x

₀

η

s₁,s₂

(m

1

)(B(0, 2))

B

0

B

^′0

η

s₁,s₂

(m

2

)(B(0, 2))

η

s₁,s₂

(m

3

)(B(0, 2))

η

s₁,s₂

(m

n

)(B(0, 2))

η

s1,s2

(m

n

)(B(0, 2))

B

n

B

_n^′

λ

n

Figure 4: Notations in the proof of Lemma 4.4 Define the homeomorphism s 4 by:

( ∀n ∈ Z >0 , s 4|B

n

= η s

1

,s

2

(m n ) ◦ g n ◦ η s

1

,s

2

(m n )

⁻¹

s 4 = Id on R ^d − S

n∈

Z>0

B n

and the homeomorphism s 5 by:

( ∀n ∈ Z >0 , s 5|B

n

= η s

1

,s

2

(m n ) ◦ h n ◦ η s

1

,s

2

(m n )

⁻¹

s 5 = Id on R ^d − S

n∈

Z>0

B n .

Let G = {s ^ǫ _i , i ∈ {1, . . . , 5} et ǫ ∈ {−1, 1}}. Let

λ n = η s

1

,s

2

(m n ) ◦ s 3 ◦ η s

1

,s

2

(m n )

⁻¹

B _n

^′

= λ n (B n )

Notice that the balls B n and B _n

^′

are disjoint and contained in η s

1

,s

2

(m n )(B(0, 2)). Notice also that the homeomorphism s 4 ◦λ n ◦ s

⁻¹

₄ ◦ λ

⁻¹

_n (respectively s 5 ◦ λ n ◦ s

⁻¹

₅ ◦ λ

⁻¹

_n , s

⁻¹

₄ ◦ s

⁻¹

₅ ◦ λ n ◦s 5 ◦s 4 ◦λ

⁻¹

_n ) fixes the points outside B n ∪ B

^′

_n , is equal to η s

1

,s

2

(m n ) ◦ g n ◦ η s

1

,s

2

(m n )

⁻¹

(respectively to η s

1

,s

2

(m n ) ◦ h n ◦ η s

1

,s

2

(m n )

⁻¹

, η s

1

,s

2

(m n ) ◦ g

⁻¹

_n ◦ h

⁻¹

_n ◦ η s

1

,s

2

(m n )

⁻¹

) on B n and to λ n ◦ η s

1

,s

2

(m n ) ◦ g

⁻¹

_n ◦ η s

1

,s

2

(m n )

⁻¹

◦ λ

⁻¹

_n (respectively to λ n ◦ η s

1

,s

2

(m n ) ◦ h

⁻¹

_n ◦ η s

1

,s

2

(m n )

⁻¹

◦ λ

⁻¹

_n , λ n ◦ η s

1

,s

2

(m n ) ◦ h n ◦ g n ◦ η s

1

,s

2

(m n )

⁻¹

◦ λ

⁻¹

_n ) on B

^′

_n .

Therefore, the homeomorphism

[s 4 , λ n ][s 5 , λ n ][s

⁻¹

₄ s

⁻¹

₅ , λ n ]

is equal to η s

1

,s

2

(m n ) ◦ f n ◦ η s

1

,s

2

(m n )

⁻¹

on B n and fixes the points outside B n . Thus:

f n = η s

1

,s

2

(m n )

⁻¹

[s 4 , λ n ][s 5 , λ n ][s

⁻¹

₄ s

⁻¹

₅ , λ n ]η s

1

,s

2

(m n ).

The homeomorphism f n hence belongs to the group generated by G and:

l

G

(f n ) ≤ 2l

G

(η s

1

,s

2

(m n )) + 6l

G

(λ n ) + 8

≤ 2l

G

(η s

1

,s

2

(m n )) + 12l

G

(η s

1

,s

2

(m n ) + 14)

≤ 14log 2 (n) + 14.

Proof of Lemma 4.5. First, let us prove the lemma in the case of homeomorphisms of R. By perturbing two given homeomorphisms (as in [12]), one can find two compactly-supported homeomorphisms ˆ s 1 and ˆ

s 2 of R which satisfy the following property:

∀m ∈ L 2 , m

^′

∈ L 2 , m 6= m

^′

⇒ η ˆ s

1

,ˆ s

2

(m)(0) 6= η s ˆ

1

,ˆ s

2

(m

^′

)(0).

Then, in the same way as in Denjoy’s construction (see [15] p.403), replace each point of the orbit of 0 under L 2 with an interval with positive length to obtain the wanted property. Thus, the proof is completed in the one-dimensional case. In the case of a higher dimension, denote by f and g the two homeomorphisms of R that we obtained in the one-dimensional case. Let [−M, M ] be an interval which contains the support of each of these homeomorphisms.

Let us look now at the case of R ^d . The homeomorphism:

R ^d → R ^d

(x 1 , x 2 , . . . , x d ) 7→ (f (x 1 ), f (x 2 ), . . . , f(x d ))

preserves the cube [−M, M ] ^d . Let s 1 be a homeomorphism of R ^d supported in [−M − 1, M + 1] ^d which is equal to the above homeomorphism on [−M, M ] ^d . Apply the same construction to the homeomorphism g to obtain a homeomorphism s 2 . The ball centered on 0 of radius 2 of R ^d is contained in the cube [−2, 2] ^d and the diameters of the sets

η s

1

,s

2

(m)([−2, 2] ^d ) = (η f,g (m)([−2, 2])) ^d

converge to 0 when the length of the word m tends to infinity. Therefore, we have the wanted property.

The case of the half-spaces H ^d is similar as long as compactly-supported homeomorphisms which are equal to homeomorphisms of the form

R + × R ^d−1 → R + × R ^d−1

(t, x 1 , x 2 , . . . , x d−1 ) 7→ ( ₂ ^t , f(x 1 ), f (x 2 ), . . . , f(x d−1 )) in a neighbourhood of 0 are used.

5 Case of surfaces with boundary

Suppose that the boundary of the surface S is nonempty. Let us prove now Proposition 3.2. By

considering a cover by half-discs, one can prove, with the same techniques as below, Proposition 3.1 in

the case that S has a nonempty boundary: this case is left to the reader.

... ...

η

s₁,s₂

(m

1

)(B(0, 2)) η

s₁,s₂

(m

2

)(B(0, 2))

η

s1,s2

(m

_n−1

)(B(0, 2)) η

s₁,s₂

(m

n

)(B(0, 2)) η

s₁,s₂

(m

n+1

)(B(0, 2))

B

1

B

1^′

B

2

B

2^′

B

_n−1

B

n

B

n+1

B

_n−1^′

B

^′_n

B

^′n+1

s ₄

g

1

g

2

g

_n−1

g

n

g

n+1

... ...

η

s₁,s₂

(m

1

)(B(0, 2)) η

s₁,s₂

(m

2

)(B (0,2))

η

s₁,s₂

(m

n−1

)(B(0, 2)) η

s₁,s₂

(m

n

)(B(0, 2)) η

s₁,s₂

(m

n+1

)(B(0, 2))

B

1

B

2

B

_n−1

B

n

B

n+1

B

^′₁

B

^′2

B

^′_n−1

B

_n^′

B

_n+1^′

λ _n s ⁻ ₄ ¹ λ ⁻ _n ¹

g

⁻₁¹

g

⁻¹2

g

⁻_n−¹1

g

⁻_n¹

g

n+1⁻¹

... ...

η

s₁,s₂

(m

1

)(B(0, 2)) η

s₁,s₂

(m

2

)(B(0, 2))

η

s₁,s₂

(m

n−1

)(B(0, 2)) η

s1,s2

(m

n

)(B(0, 2)) η

s₁,s₂

(m

n+1

)(B(0, 2))

B

1

B

₁^′

B

2

B

2^′

B

_n−1

B

n

B

n+1

B

_n−1^′

B

^′_n

B

^′_n+1

[s ₄ , λ _n ]

g

n

g

⁻¹_n

... ...

η

s₁,s₂

(m

1

)(B(0, 2)) η

s₁,s₂

(m

2

)(B(0, 2))

η

s₁,s₂

(m

_n−1

)(B(0, 2)) η

s₁,s₂

(m

n

)(B(0, 2)) η

s₁,s₂

(m

n+1

)(B(0, 2))

B

1

B

1^′

B

2

B

₂^′

B

n−1

B

n

B

n+1

B

_n−^′ 1

B

^′_n

B

^′n+1

[ s ₅ , λ _n ]

h

n

h

⁻_n¹

... ...

η

s₁,s₂

(m

1

)(B(0, 2)) η

s₁,s₂

(m

2

)(B(0, 2))

η

s₁,s₂

(m

n−1

)(B(0, 2)) η

s1,s2

(m

n

)(B (0, 2)) η

s₁,s₂

(m

n+1

)(B(0, 2))

B

1

B

^′₁

B

2

B

2^′

B

_n−1

B

n

B

n+1

B

^′_n−1

B

_n^′

B

^′_n+1

[s ⁻ ₄ ¹ s ⁻ ₅ ¹ , λ _n ]

g

⁻_n¹

h

⁻_n¹

h

n

g

n

... ...

η

s₁,s₂

(m

1

)(B(0, 2)) η

s₁,s₂

(m

2

)(B(0, 2))

η

s₁,s₂

(m

n−1

)(B(0, 2)) η

s₁,s₂

(m

n

)(B(0, 2)) η

s₁,s₂

(m

n+1

)(B(0, 2))

B

1

B

2

B

_n−1

B

n

B

n+1

B

^′1

B

^′2

B

^′_n−1

B

_n^′

B

n+1^′

[ s ₄ , λ _n ][ s ₅ , λ _n ][ s ⁻ ₄ ¹ s ⁻ ₅ ¹ , λ _n ]

[g

n

, h

n

]

Figure 5: The different homeomorphisms appearing in the proof of Lemma 4.4

Recall that, in Section 3, we have chosen a "nice" fundamental domain D 0 . Let A ˜ be the set of edges of the boundary ∂D 0 which are not contained in the boundary of S ˜ and let:

A = n

Π(β), β ∈ A ˜ o .

For any edge α in A, let us consider a closed disc V α which does not meet the boundary of the surface S, whose interior contains α ∩ S

^′

and such that there exists a homeomorphism ϕ α : V α → D ² which sends the set α ∩ V α to the horizontal diameter of the unit disc D ² . Choose sufficiently thin discs V α so that they are pairwise disjoint. Let U 1 be a closed disc which contains the union of the discs V α . Let U 2 be a closed disc of S which does not meet any edge in A, i.e. contained in the interior of the fundamental domain D 0 , and which satisfies the two following properties:

1. The surface S

^′

is contained in the interior of S

α∈A

V α ∪ U 2 .

2. For any edge α in A, the set U 2 ∩ V α is homeomorphic to the disjoint union of two closed discs.

Let U = {U 1 , U 2 }.

∂ S ˜

∂ S ˜

∂ S ˜

∂ S ˜

U

₂

V

_α

1

V

_α

2

V

_α

1

V

_α

2

Figure 6: Notations in the case of surfaces with boundary

The proof of the inequality in the case of the group Homeo 0 (S, ∂S) requires the following lemmas:

Lemma 5.1. Let g be a homeomorphism in Homeo 0 (S, ∂S) supported in the interior of S V α ∪U 2 . Suppose that el D

0

(˜ g(D 0 )) ≥ 2. Then there exist homeomorphisms g 1 , g 2 and g 3 in Homeo 0 (S, ∂S) supported respectively in the interior of S V α , U 2 and S V α such that the following property is satisfied:

el D

0

(˜ g 3 ◦ g ˜ 2 ◦ ˜ g 1 ◦ ˜ g(D 0 )) ≤ el D

0

(˜ g(D 0 )) − 1.

Lemma 5.2. Let g be a homeomorphism in Homeo 0 (S, ∂S) supported in the interior of S

V α ∪ U 2 . If el D

0

(˜ g(D 0 )) = 1, then:

Frag

_U

(g) ≤ 6.

End of the proof of Proposition 3.2. Let k = el D

0

(˜ g(D 0 )). By Lemma 5.1, after composing ˜ g with 3(k− 1) homeomorphisms, each supported in one of the discs of U , we obtain a homeomorphism f 1 supported in

S

α∈A

V α ∪ U 2 with:

el D

0

( ˜ f 1 (D 0 )) = 1.

Then, apply Lemma 5.2 to the homeomorphism f 1 :

Frag

_U

(f 1 ) ≤ 6.

Therefore:

Frag

_U

(g) ≤ 3(el D

0

(˜ g(D 0 )) − 1) + 6.

However, as D 0 ∩ g(D ˜ 0 ) 6= ∅ (the homeomorphism g pointwise fixes a neighbourhood of the boundary of S):

el D

0

(˜ g(D 0 )) ≤ diam

D

(˜ g(D 0 )).

Hence:

Frag

_U

(g) ≤ 3diam

D

(˜ g(D 0 )) + 3.

Notice that we indeed proved the following more precise proposition:

Proposition 5.3. Let g be a homeomorphism in Homeo 0 (S, ∂S) supported in the interior of S

α∈A

V α ∪ U 2 . Then:

Frag

_U

(g) ≤ 3diam

D

(˜ g(D 0 )) + 3.

Proof of Lemma 5.1. Let us first give the properties of the homeomorphisms g 1 , g 2 and g 3 which will satisfy the conclusion of the lemma. Let us give an idea of the action of these homeomorphisms "with the hands".

If we look at the pieces of the disc g(D ˜ 0 ) furthest from D 0 , the homeomorphism g 1 repulses them back to the open set U 2 , the homeomorphism g 2 repulses them outside the open set U 2 and the homeomorphism g 3 makes them exit from the fundamental domain of D in which these pieces were contained (see figure 7). Let us give the precise construction of these homeomorphisms.

∂ S ˜

∂ S ˜

∂ S ˜

∂ S ˜

U

2

˜ g(∂D

0

)

g

1

g

2

g

2

g

3

g

3

g

1

g

2

Figure 7: Illustration of the proof of Lemma 5.1 Let g 1 be a homeomorphism supported in S

α∈A

V α such that:

1. The homeomorphism g 1 pointwise fixes Π(∂D 0 ).

2. For any edge α and any connected component C of V α ∩ g(Π(∂D 0 )) which does not meet Π(∂D 0 ):

g 1 (C) ⊂ U 2 .